Kapitza Inverted Pendulum

A pendulum standing upside-down, and staying there. Its pivot is shaken straight up and down, $y_p(t) = a\cos(\omega t)$, fast enough that the rapid jitter averages into a restoring force toward the top. Above the Kapitza threshold $a^2\omega^2 > 2gl$ the inverted position is genuinely stable: nudge it and it wobbles back upright. The trick shows up cleanly in the time-averaged effective potential $U_\text{eff}(\theta) = mgl\cos\theta + \tfrac14 m(a\omega)^2\sin^2\theta$, which (above threshold) grows a small well at the top, $\theta = 0$. The scene shows the shaken pendulum holding upright; the diagnostic plots that effective potential with a marker showing the pendulum sitting in its well.

Figure 1. Kapitza inverted pendulum. Top: the rod held upright by a vertically vibrated pivot. Bottom: the time-averaged effective potential versus angle from vertical, with a marker at the current state sitting in the well at the top once stable. Method: RK4 on the driven equation of motion.

a (m)0.120

omega55

tilt (deg)22

WHAT TO TRY

Start at the default: the rod is upside-down and stays there, jittering. The effective potential below shows a well at the top.
Lower the amplitude or frequency until the ratio drops below 1: the well at the top flips to a hump and the pendulum falls over.
Raise the frequency back up and the upright re-stabilizes. The threshold for the equilibrium is exactly a²ω² = 2gl, but that is a leading-order result: it guarantees small wobbles stay, not that any tilt survives.
Sit just above the threshold and increase the starting tilt: the gauge shows the criterion met, yet the rod topples. Near the line the basin of attraction is tiny, so only small tilts are caught; push the ratio well past 1 and even large tilts are recovered.